One of the major ideas of Shafer's mathematical theory of evidence is the introduction of uncertainty descriptions on different representation domains of phenomena, called families of compatible frames of discernment. Here we are going to analyze these families of frames from an algebraic point of view, study the properties of minimal refinements of collections of domains and introduce the internal operation of maximal coarsening to establish the structure of semimodular lattice. Motivated by the search for a solution of the conflict problem that arises in sensor fusion applications, we will show the connection between classical independence of frames as Boolean subalgebras and independence of frames as elements of a locally finite Birkhoff lattice. This will eventually suggest a potential algebraic solution of the conflict problem.
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References
A. Beutelspacher and U. Rosenbaum, Projective Geometry (Cambridge University Press, Cambridge, 1998).
G. Birkhoff, Abstract linear dependence and lattices, American Journal of Mathematics 57 (1935) 800–804.
G. Birkhoff, Lattice Theory (3rd ed.) Vol. 25 (Amer. Math. Soc. Colloquium Publications, Providence, Rhode Island, 1967).
F. Cuzzolin, Visions of a generalized probability theory. PhD dissertation, Università di Padova, Dipartimento di Elettronica e Informatica (2001).
F. Cuzzolin and R. Frezza, An evidential reasoning frame-work for object tracking, in: SPIE – Photonics East 99, Vol. 3840 (19–22 September 1999) pp. 13–24.
F. Cuzzolin and R. Frezza, Integrating feature spaces for object tracking, in: Proc. of MTNS2000 (21–25 June 2000).
A. Dempster, Upper and lower probabilities generated by a random closed interval, Annals of Mathematical Statistics 39 (1968) 957–966.
A. Dempster, Upper and lower probabilities inferences for families of hypothesis with monotone density ratios, Annals of Mathematical Statistics 40 (1969) 953–969.
A.P. Dempster, Upper and lower probability inferences based on a sample from a finite univariate population, Biometrika 54 (1967) 515–528.
M. Deutsch-McLeish, A study of probabilities and belief functions under conflicting evidence: Comparisons and new method, in: Proceedings of IPMU'90 (Paris, France, 2–6 July 1990) pp. 41–49.
I.R. Goodman and H.T. Nguyen, Uncertainty Models for Knowledge-Based Systems (North Holland, New York, 1985).
J. Goutsias, R.P. Mahler and H.T. Nguyen, Random sets: theory and applications, in: IMA Volumes in Mathematics and Its Applications, Vol. 97 (Springer, Berlin Heidelberg New York, December 1997).
N. Jacobson, Basic Algebra I (Freeman, New York, 1985).
J. Kohlas and Monney P.-A., A Mathematical Theory of Hints – An Approach to the Dempster–Shafer Theory of Evidence. Lecture Notes in Economics and Mathematical Systems, (Springer, 1995).
G. Matheron, Random Sets and Integral Geometry, Wiley Series in Probability and Mathematical Statistics.
C.K. Murphy, Combining belief functions when evidence conflicts, Decision Support Systems 29 (2000) 1–9.
H. Nguyen, On random sets and belief functions, J. Mathematical Analysis and Applications 65 (1978) 531–542.
H. Nguyen and T. Wang, Belief functions and random sets, in: Applications and Theory of Random Sets, The IMA Volumes in Mathematics and its Applications, Vol. 97 (Springer, 1997) pp. 243–255.
J.G. Oxley, Matroid Theory (Oxford University Press, Great Clarendon Street, Oxford, UK, 1992).
K.I. Rosenthal, Quantales and Their Applications (Longman scientific and technical, Longman house, Burnt Mill, Harlow, Essex, UK, 1990).
G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, 1976).
G. Shafer, P.P. Shenoy and K. Mellouli, Propagating belief functions in qualitative Markov trees, International Journal of Approximate Reasoning 1(4) (1987) 349–400.
R. Sikorski, Boolean Algebras (Springer, Berlin Heidelberg New York, 1964).
P. Smets, The transferable belief model and random sets, International Journal of Intelligent Systems 7 (1992) 37–46.
M. Stern, Semimodular Lattices (Cambridge University Press, 1999).
G. Szasz, Introduction to Lattice Theory (Academic, New York, 1963).
H. Whitney, On the abstract properties of linear dependence, American Journal of Mathematics 57 (1935) 509–533.
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Cuzzolin, F. Algebraic structure of the families of compatible frames of discernment. Ann Math Artif Intell 45, 241–274 (2005). https://doi.org/10.1007/s10472-005-9010-1
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DOI: https://doi.org/10.1007/s10472-005-9010-1